Gamma Analytical Modeling Evolution (GAME) I: The physical implications of deriving the stellar mass functions from z=0 to z=8

The $\Gamma$ growth model is an effective parameterization employed across various scientific disciplines and scales to depict growth. It has been demonstrated that the cosmic star formation rate density (CSFRD) can also be described broadly by this pattern, i.e. $\frac{dM(T)}{dT} = M_{z,0}\, \times \frac{\beta^{\alpha}}{\Gamma(\alpha)} \, T^{\alpha-1} e^{-\beta \, T }$ M$_{\odot}$ Gyr$^{-1}$, where $M_{z,0}$ is the stellar mass at $z$ = 0, $\alpha = 3.0$, $\beta = 0.5 $ Gyr$^{-1}$ and $T$ describes time. We use the identical $\Gamma$ growth pattern given by the CSFRD to extend the present day (z = 0) stellar mass bins $M_{\ast}(T)$ of the Galaxy Stellar Mass Function (GSMF) and investigate if we are able to reproduce observations for the high redshift GSMFs. Surprisingly, our scheme describes successfully the evolution of the GSMF over 13.5 Gyrs, especially for objects with intermediate and low masses. We observe some deviations that manifest {\it solely} at very high redshifts ($z > 1.5$, i.e. more than 9.5 Gyr ago) and {\it specifically} for very small and exceedingly massive objects. We discuss the possible solutions (e.g. impacts of mergers) for these offsets. Our formalism suggests that the evolution of the GSMF is set by simple (few parameters) and physically motivated arguments. The parameters $\beta$ and $\alpha$ are theoretically consistent within a multi-scale context and are determined from the dynamical time scale ($\beta$) and the radial distribution of the accreting matter ($\alpha$). We demonstrate that both our formalism and state-of-the-art simulations are consistent with recent GSMFs derived from JWST data at high redshifts.